It was the third consecutive race win for Bagnaia following his victories at the British GP and the Dutch GP. The Italian finished 0.492 seconds ahead of Quartararo, while Jack Miller completed the podium.
He became only the second Italian rider in the modern MotoGP era to win three consecutive races after Valentino Rossi, who retired last year after winning nine world titles.
"It was a long race... I did too many mistakes in the first part of the year, so it was time for me to be more smart," Bagnaia said in a post-race interview.
"When I looked at the gap, I tried to be very cautious with the time laps because I was sure that my pace was good enough to open the gap and in the last two laps, I just tried to be calm. I am very happy with my team, they have done an incredible job."
Starting second on the grid, Bagnaia made a lightning start at the Red Bull Ring as he overtook pole sitter Enea Bastianini of Gresini Racing on the opening lap.
Bastianini dropped further down to fourth on the sixth lap when Bagnaia's team mate Miller and Pramac Ducati's Jorge Martin moved up before he was forced to end his race due to a deflating front tyre after running his bike into the gravel.
Martin had looked set for a podium finish but he ran wide at the chicane on the 17th lap, surrendering third spot to Yamaha rider Quartararo, who started fifth on the grid.
Quartararo then piled pressure on Miller for second and succeeded with less than three laps to go after pulling off a stunning move in the middle of the chicane at turn two.
The Frenchman now has a 32-point lead over Aprilia's Aleix Espargaro in the riders championship, while Bagnaia is third, 44 points behind the leader.
